# Thread: Finding the length of part of a curve.

1. ## Finding the length of part of a curve.

Like the title of this thread suggests, I need to find the length of a section of the curve. I've done most of the questions but I got stuck on these ones:

I'm clearly doing the polar co-ordinate ones incorrectly, and the parametric one has me slightly confused (are the limits t=at^2 and t=0?)

If anyone can solve any of these it would be a great help!

If anyone wants I can post my work so far. It's just really long!

NOTE THAT 7 HAS BEEN DONE! =D

2. Here's my working for the first one (ie.question 7):

I can't really see what's wrong with my answer. It's different to the one in the book so I must be making a mistake somewhere.

3. the limits for the first one should be from t=0 to t=t

because when t=t, x=at^2 and y=2at which is what is asked

4. Ohhh!! I get it!

Is the rest of my working fine though?

5. The book gives the answer as:

I'm not really sure where the last part comes from. It looks a little bit like an arsinh.

6. theres a problem with your substitution,

make the substitution
$t=\sinh u$

the integral then reduces to
$2a \int_{0}^{\sinh^{-1}t} \cosh^2 u du$

calculating that gets you to the answer

$a(\sinh^{-1}t+t\sqrt{t^2+1})$

7. Just to add for the polar ones have a look here
Pauls Online Notes : Calculus II - Arc Length with Polar Coordinates

theres a good derivation and the example given is pretty much Q11 for you

8. did you work it all the way through?

The second part to get the t(t^2+1)^0.5 is taking me ages!

9. I've done the integration and i've worked through it but i've hit a snag:

Is there an easier way to work this out? I've practically covered my whiteboard in working and i'm not any closer to getting what I need to.

My integration went like:

10. Originally Posted by Showcase_22
I've done the integration and i've worked through it but i've hit a snag:

Is there an easier way to work this out? I've practically covered my whiteboard in working and i'm not any closer to getting what I need to.
I just did it the lazy way without switching to exponentials

use $\cosh x = \sqrt{\sinh^2 x +1}$

so $\cosh (\sinh^{-1}t)=\sqrt{(\sinh (\sinh^{-1}t))^2+1}=\sqrt{t^2+1}$

11. lol! That's waaaaaay easier than doing it the way I was.

Thanks.

I'm working on question 11 now. I was able to get an answer for it before, but it was different to the one in the back of the book so i'll read what's at the address you posted and try again.

12. All that for the wrong answer.

Apparently the answer is just 8a.

???